Image of a linear transformation matrix...

I know I've already posted a number of threads about this, but I still can't get my head around it. (Headbang)

I get how to get the Kernel of the matrix associated with a transformation.

But for some reason getting the image, which seems to be very closely linked, I just can't understand.

The example 3.58 in this link: Linear maps - HowTo: Kernels and Images - HowTo: Kernels and Images III seems to be the most straightforward of all the 'instructions' I've come across.

So the first step is fine.. then "hence we see that we must extend to a basis for R^5 by adding the elementary vectors e_3, e_4, e_5..."

To "whence" does he refer? How does one 'extend to a basis', and how does one 'add' the elementary vectors - whatever they are?

This is holding me back again and again, and it's infuriating because I know it should be simple! (Angry)

Thanks in advance for any help..